The present invention relates generally to deformation analysis and, more particularly, to finite element mesh processes used in deformation analysis.
The process of finite element analysis that deals with large deformation usually produces distorted elements at the later stages of the analysis. These distorted elements lead to several problems; inaccurate results, slow convergence and premature analysis termination. Metal-forming processes are the most common applications involved with large deformation analysis; they include forging, extrusion, rolling, deep drawing, and so on. An example of such large deformation analysis is illustrated in FIGS. 1a-b. This is a three-dimensional forging example in which a sinusoidal die (not shown) deforms a deformable blank 5 into a geometry with high-curvature corners. As the finite element analysis is performed on this problem using a pure Lagrangian method (shown in FIG. 1b), several elements are severely distorted especially around high-curvature corners. Consequently, as can be seen in FIG. 1b, the resulting mesh contains many highly-distorted elements and inverted elements at the later stages, potentially leading to the several problems listed earlier.
There are two conventional techniques for addressing this problem, the adaptive remeshing and the Arbitrary Lagrangian-Eulerian (ALE) techniques. Both techniques, however, have drawbacks.
Adaptive remeshing is a technique which replaces an over-distorted mesh with a better-conditioned mesh when the error approximation of analysis exceeds the tolerance, or the maximum error value allowed. The newly-created mesh may not necessarily have the same topology as the original mesh, and the number of nodes and elements of the new mesh may differ from the original mesh. Therefore, state variables and history-dependent variables must also be transferred from the original to the new mesh. State variables include nodal displacements and variables of the contact algorithm. History-dependent variables are the stress tensor, strain tensor, plastic strain tensor, etc. The adaptive remeshing technique usually completely remeshes the part at every certain number of steps in the analysis. The disadvantage of this method is its high computational cost, especially during the procedure for determining the error estimator and mapping variables from an old to a new mesh. More importantly, computational costs increase considerably for analysis of complicated geometries.
The Arbitrary Lagrangian-Eulerian (ALE) method is another technique for addressing the problem of large deformation in finite element analysis. This method combines the features of pure Lagrangian analysis and Eulerian analysis—two common types of finite element analysis. In pure Lagrangian analysis, a mesh follows the material deformation during analysis; the mesh is connected to the material throughout the finite element calculation. Since the mesh and the material are connected, severe distortion of the mesh can cause difficulty in the finite element calculation. It is here that adaptive remeshing must be applied to improve the shape quality of the mesh in order to continue the analysis. ALE was developed to reduce the repetition of complete remeshing. Essentially, ALE is a Lagrangian analysis that takes advantage of the advection techniques of Eulerian analysis. In the ALE method, the mesh is neither connected to the material nor fixed to a spatial coordinate system. Rather, it is prescribed in an arbitrary manner. During the analysis, the mesh elements deform according to the Lagrangian method. However, instead of repositioning the nodes at their original position and performing advection, as in the Eulerian method, the nodes are placed at other positions to obtain optimal mesh quality. The mesh nodes have velocity associated with them in order to obtain the updated mesh. Mesh velocity plays an important role in the ALE method as it reduces the analysis error and prevents mesh distortion. Another important characteristic of this method is that it changes the location of the nodes in the existing mesh, instead of creating a completely new mesh, like the adaptive remeshing method, and it maintains the same (or similar) mesh topology throughout the analysis. However, because of its complexity, the computation cost is much more expensive than using pure Lagrangian analysis. There are also other limitations in ALE analysis. In many cases the mesh suffers considerable distortion and the same mesh topology cannot be maintained for the entire analysis. In such cases, complete adaptive remeshing is still required. Another drawback of ALE is that the state-variables remapping step is much more complicated than in the Lagrangian method.